From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Monday Sept. 11, 2000,
10:02PM
Consider the fixed point g(x,y) = (x,y) for g(x,y) = (x-->y, (x-->y)^)
where x-->y is 1 - x + y and its real conjugate, labeled (x-->y)^, is 1 +
x - y. This is equivalent to 1 - x + y = x, 1 - y + x = y, and subtracting
the second equation from the first results in -x + y + y - x = x - y or
3(y - x) = 0 so y = x and substituting for either x or y in the first or
second equation yields the unique solution x = 1, y = 1, so the fixed point
is (1, 1), and since x-->y = x and y-->x = y, we also have x-->y = 1 and
y-->x = 1. It turns out that even if the problem had been g(x,y) = (y,x)
for the above g, the same fixed point would have been obtained.
This is remarkable for those interested in fixed points. Fixed points have
turned out to be of considerable importance across categories, and similarly
for symmetry groups and invariance properties, both in mathematics and
physics. From the above paragraph, the multiplicative unit of the category
has turned out to be its unique fixed point. Also it suggests that
categories for which the multiplicative unit is very important are those in
which x-->y = 1 - x + y plays an important role. This holds for at least
two categories, which I will refer to for short as probabilities and (fuzzy)
multivalued logic - especially Product/Goguen logic and Lukaciewicz logic,
although both are related to Godel logic. The multiplicative unit is
important in probability because probabilities by definition are between 0
and 1 - their multiplicative unit is also their maximum. In (fuzzy)
multivalued logics, the unit 1 is similarly an extreme value, although it is
usually considered to be the "trivial" case of either tautology or
contradiction (usually tautology). This would at least preserve the
intuitive idea of tautology expressing "complete logical truth", however
trivial we may consider that to be.
The above results also suggest that a generalization of fixed point results
to the general case where x-->y and/or y-->x = 1 might be quite useful, and
this is precisely what logic-based probability (LBP) has found to be the
case (recall my previous contributions to categories@mta.ca).
There is an interesting application to history that I might cite as an
amusing aside. History is usually taught in terms of stories - the story
of a war, a civilization, an era, etc. The above results would suggest, if
they are applicable to history, that it is best to teach history as the
study of rare discrete events and how they causally influence each other.
For example, what caused a war, what ended a war, or in different language,
why did the war occur, why did the war end, etc. The discrete events would
correspond to fixed points, and we would be especially interested in them
when they are repeated in history - the repetition would correspond to fixed
points or invariance in time or symmetry in time. Then history would become
the study of why errors repeat rather than merely the study of what
happened, where it happened, who it happened to, how it happened, etc. I
find this somewhat amusing because of arguments that one can get into when
discussing matters with the more detail-obsessed historians to whom any
suggestion of causation or relationships across categories results sometimes
in remarkable responses.
Osher Doctorow